Let $f:M\rightarrow N$ be a smooth map; surjective submersion and all that is necessary to call it a well behaved map.
Let $\Gamma$ be a connection on $N$ (a connection on the tangent bundle $TN\rightarrow N$).
Can I talk about the pull back of connection $\Gamma$ along $f:M\rightarrow N$ to get a connection on $M$?
I do not mean to ask for connection on the vector bundle obtained by pullback of vector bundle $TN\rightarrow N$ along the map $f:M\rightarrow N$ namely the vector bundle $M\times_N TN\rightarrow M$. No, I am not asking about connection on this vector bundle $M\times_NTN\rightarrow M$. I am asking for connection on the tangent bundle $TM\rightarrow M$.
Given a connection on tangent bundle $TN\rightarrow N$, can I get a connection on tangent bundle $TM\rightarrow M$ obtained by pull back of connection on $TN\rightarrow N$ along the map $f:M\rightarrow N$?
You can "twist" the notion of "pullback" slightly. You can assume $f$ is well behaved except that it is a diffeomorphism.